direct product, metacyclic, supersoluble, monomial, 2-hyperelementary
Aliases: C3×C8⋊D7, C24⋊7D7, C168⋊9C2, C56⋊12C6, C21⋊5M4(2), D14.3C12, C12.57D14, C84.65C22, Dic7.3C12, C7⋊C8⋊11C6, C8⋊3(C3×D7), (C4×D7).5C6, (C6×D7).3C4, C2.3(C12×D7), C6.16(C4×D7), C4.13(C6×D7), C7⋊4(C3×M4(2)), C42.20(C2×C4), C28.37(C2×C6), (C12×D7).5C2, C14.16(C2×C12), (C3×Dic7).3C4, (C3×C7⋊C8)⋊11C2, SmallGroup(336,59)
Series: Derived ►Chief ►Lower central ►Upper central
Generators and relations for C3×C8⋊D7
G = < a,b,c,d | a3=b8=c7=d2=1, ab=ba, ac=ca, ad=da, bc=cb, dbd=b5, dcd=c-1 >
Smallest permutation representation of C3×C8⋊D7
►On 168 points
Generators in S168
(1 132 165)(2 133 166)(3 134 167)(4 135 168)(5 136 161)(6 129 162)(7 130 163)(8 131 164)(9 116 64)(10 117 57)(11 118 58)(12 119 59)(13 120 60)(14 113 61)(15 114 62)(16 115 63)(17 124 65)(18 125 66)(19 126 67)(20 127 68)(21 128 69)(22 121 70)(23 122 71)(24 123 72)(25 138 88)(26 139 81)(27 140 82)(28 141 83)(29 142 84)(30 143 85)(31 144 86)(32 137 87)(33 73 96)(34 74 89)(35 75 90)(36 76 91)(37 77 92)(38 78 93)(39 79 94)(40 80 95)(41 152 104)(42 145 97)(43 146 98)(44 147 99)(45 148 100)(46 149 101)(47 150 102)(48 151 103)(49 160 112)(50 153 105)(51 154 106)(52 155 107)(53 156 108)(54 157 109)(55 158 110)(56 159 111)
(1 2 3 4 5 6 7 8)(9 10 11 12 13 14 15 16)(17 18 19 20 21 22 23 24)(25 26 27 28 29 30 31 32)(33 34 35 36 37 38 39 40)(41 42 43 44 45 46 47 48)(49 50 51 52 53 54 55 56)(57 58 59 60 61 62 63 64)(65 66 67 68 69 70 71 72)(73 74 75 76 77 78 79 80)(81 82 83 84 85 86 87 88)(89 90 91 92 93 94 95 96)(97 98 99 100 101 102 103 104)(105 106 107 108 109 110 111 112)(113 114 115 116 117 118 119 120)(121 122 123 124 125 126 127 128)(129 130 131 132 133 134 135 136)(137 138 139 140 141 142 143 144)(145 146 147 148 149 150 151 152)(153 154 155 156 157 158 159 160)(161 162 163 164 165 166 167 168)
(1 25 113 148 74 65 153)(2 26 114 149 75 66 154)(3 27 115 150 76 67 155)(4 28 116 151 77 68 156)(5 29 117 152 78 69 157)(6 30 118 145 79 70 158)(7 31 119 146 80 71 159)(8 32 120 147 73 72 160)(9 48 37 127 53 168 83)(10 41 38 128 54 161 84)(11 42 39 121 55 162 85)(12 43 40 122 56 163 86)(13 44 33 123 49 164 87)(14 45 34 124 50 165 88)(15 46 35 125 51 166 81)(16 47 36 126 52 167 82)(17 105 132 138 61 100 89)(18 106 133 139 62 101 90)(19 107 134 140 63 102 91)(20 108 135 141 64 103 92)(21 109 136 142 57 104 93)(22 110 129 143 58 97 94)(23 111 130 144 59 98 95)(24 112 131 137 60 99 96)
(1 153)(2 158)(3 155)(4 160)(5 157)(6 154)(7 159)(8 156)(9 33)(10 38)(11 35)(12 40)(13 37)(14 34)(15 39)(16 36)(17 138)(18 143)(19 140)(20 137)(21 142)(22 139)(23 144)(24 141)(25 65)(26 70)(27 67)(28 72)(29 69)(30 66)(31 71)(32 68)(42 46)(44 48)(49 168)(50 165)(51 162)(52 167)(53 164)(54 161)(55 166)(56 163)(57 93)(58 90)(59 95)(60 92)(61 89)(62 94)(63 91)(64 96)(73 116)(74 113)(75 118)(76 115)(77 120)(78 117)(79 114)(80 119)(81 121)(82 126)(83 123)(84 128)(85 125)(86 122)(87 127)(88 124)(97 101)(99 103)(105 132)(106 129)(107 134)(108 131)(109 136)(110 133)(111 130)(112 135)(145 149)(147 151)
G:=sub<Sym(168)| (1,132,165)(2,133,166)(3,134,167)(4,135,168)(5,136,161)(6,129,162)(7,130,163)(8,131,164)(9,116,64)(10,117,57)(11,118,58)(12,119,59)(13,120,60)(14,113,61)(15,114,62)(16,115,63)(17,124,65)(18,125,66)(19,126,67)(20,127,68)(21,128,69)(22,121,70)(23,122,71)(24,123,72)(25,138,88)(26,139,81)(27,140,82)(28,141,83)(29,142,84)(30,143,85)(31,144,86)(32,137,87)(33,73,96)(34,74,89)(35,75,90)(36,76,91)(37,77,92)(38,78,93)(39,79,94)(40,80,95)(41,152,104)(42,145,97)(43,146,98)(44,147,99)(45,148,100)(46,149,101)(47,150,102)(48,151,103)(49,160,112)(50,153,105)(51,154,106)(52,155,107)(53,156,108)(54,157,109)(55,158,110)(56,159,111), (1,2,3,4,5,6,7,8)(9,10,11,12,13,14,15,16)(17,18,19,20,21,22,23,24)(25,26,27,28,29,30,31,32)(33,34,35,36,37,38,39,40)(41,42,43,44,45,46,47,48)(49,50,51,52,53,54,55,56)(57,58,59,60,61,62,63,64)(65,66,67,68,69,70,71,72)(73,74,75,76,77,78,79,80)(81,82,83,84,85,86,87,88)(89,90,91,92,93,94,95,96)(97,98,99,100,101,102,103,104)(105,106,107,108,109,110,111,112)(113,114,115,116,117,118,119,120)(121,122,123,124,125,126,127,128)(129,130,131,132,133,134,135,136)(137,138,139,140,141,142,143,144)(145,146,147,148,149,150,151,152)(153,154,155,156,157,158,159,160)(161,162,163,164,165,166,167,168), (1,25,113,148,74,65,153)(2,26,114,149,75,66,154)(3,27,115,150,76,67,155)(4,28,116,151,77,68,156)(5,29,117,152,78,69,157)(6,30,118,145,79,70,158)(7,31,119,146,80,71,159)(8,32,120,147,73,72,160)(9,48,37,127,53,168,83)(10,41,38,128,54,161,84)(11,42,39,121,55,162,85)(12,43,40,122,56,163,86)(13,44,33,123,49,164,87)(14,45,34,124,50,165,88)(15,46,35,125,51,166,81)(16,47,36,126,52,167,82)(17,105,132,138,61,100,89)(18,106,133,139,62,101,90)(19,107,134,140,63,102,91)(20,108,135,141,64,103,92)(21,109,136,142,57,104,93)(22,110,129,143,58,97,94)(23,111,130,144,59,98,95)(24,112,131,137,60,99,96), (1,153)(2,158)(3,155)(4,160)(5,157)(6,154)(7,159)(8,156)(9,33)(10,38)(11,35)(12,40)(13,37)(14,34)(15,39)(16,36)(17,138)(18,143)(19,140)(20,137)(21,142)(22,139)(23,144)(24,141)(25,65)(26,70)(27,67)(28,72)(29,69)(30,66)(31,71)(32,68)(42,46)(44,48)(49,168)(50,165)(51,162)(52,167)(53,164)(54,161)(55,166)(56,163)(57,93)(58,90)(59,95)(60,92)(61,89)(62,94)(63,91)(64,96)(73,116)(74,113)(75,118)(76,115)(77,120)(78,117)(79,114)(80,119)(81,121)(82,126)(83,123)(84,128)(85,125)(86,122)(87,127)(88,124)(97,101)(99,103)(105,132)(106,129)(107,134)(108,131)(109,136)(110,133)(111,130)(112,135)(145,149)(147,151)>;
G:=Group( (1,132,165)(2,133,166)(3,134,167)(4,135,168)(5,136,161)(6,129,162)(7,130,163)(8,131,164)(9,116,64)(10,117,57)(11,118,58)(12,119,59)(13,120,60)(14,113,61)(15,114,62)(16,115,63)(17,124,65)(18,125,66)(19,126,67)(20,127,68)(21,128,69)(22,121,70)(23,122,71)(24,123,72)(25,138,88)(26,139,81)(27,140,82)(28,141,83)(29,142,84)(30,143,85)(31,144,86)(32,137,87)(33,73,96)(34,74,89)(35,75,90)(36,76,91)(37,77,92)(38,78,93)(39,79,94)(40,80,95)(41,152,104)(42,145,97)(43,146,98)(44,147,99)(45,148,100)(46,149,101)(47,150,102)(48,151,103)(49,160,112)(50,153,105)(51,154,106)(52,155,107)(53,156,108)(54,157,109)(55,158,110)(56,159,111), (1,2,3,4,5,6,7,8)(9,10,11,12,13,14,15,16)(17,18,19,20,21,22,23,24)(25,26,27,28,29,30,31,32)(33,34,35,36,37,38,39,40)(41,42,43,44,45,46,47,48)(49,50,51,52,53,54,55,56)(57,58,59,60,61,62,63,64)(65,66,67,68,69,70,71,72)(73,74,75,76,77,78,79,80)(81,82,83,84,85,86,87,88)(89,90,91,92,93,94,95,96)(97,98,99,100,101,102,103,104)(105,106,107,108,109,110,111,112)(113,114,115,116,117,118,119,120)(121,122,123,124,125,126,127,128)(129,130,131,132,133,134,135,136)(137,138,139,140,141,142,143,144)(145,146,147,148,149,150,151,152)(153,154,155,156,157,158,159,160)(161,162,163,164,165,166,167,168), (1,25,113,148,74,65,153)(2,26,114,149,75,66,154)(3,27,115,150,76,67,155)(4,28,116,151,77,68,156)(5,29,117,152,78,69,157)(6,30,118,145,79,70,158)(7,31,119,146,80,71,159)(8,32,120,147,73,72,160)(9,48,37,127,53,168,83)(10,41,38,128,54,161,84)(11,42,39,121,55,162,85)(12,43,40,122,56,163,86)(13,44,33,123,49,164,87)(14,45,34,124,50,165,88)(15,46,35,125,51,166,81)(16,47,36,126,52,167,82)(17,105,132,138,61,100,89)(18,106,133,139,62,101,90)(19,107,134,140,63,102,91)(20,108,135,141,64,103,92)(21,109,136,142,57,104,93)(22,110,129,143,58,97,94)(23,111,130,144,59,98,95)(24,112,131,137,60,99,96), (1,153)(2,158)(3,155)(4,160)(5,157)(6,154)(7,159)(8,156)(9,33)(10,38)(11,35)(12,40)(13,37)(14,34)(15,39)(16,36)(17,138)(18,143)(19,140)(20,137)(21,142)(22,139)(23,144)(24,141)(25,65)(26,70)(27,67)(28,72)(29,69)(30,66)(31,71)(32,68)(42,46)(44,48)(49,168)(50,165)(51,162)(52,167)(53,164)(54,161)(55,166)(56,163)(57,93)(58,90)(59,95)(60,92)(61,89)(62,94)(63,91)(64,96)(73,116)(74,113)(75,118)(76,115)(77,120)(78,117)(79,114)(80,119)(81,121)(82,126)(83,123)(84,128)(85,125)(86,122)(87,127)(88,124)(97,101)(99,103)(105,132)(106,129)(107,134)(108,131)(109,136)(110,133)(111,130)(112,135)(145,149)(147,151) );
G=PermutationGroup([[(1,132,165),(2,133,166),(3,134,167),(4,135,168),(5,136,161),(6,129,162),(7,130,163),(8,131,164),(9,116,64),(10,117,57),(11,118,58),(12,119,59),(13,120,60),(14,113,61),(15,114,62),(16,115,63),(17,124,65),(18,125,66),(19,126,67),(20,127,68),(21,128,69),(22,121,70),(23,122,71),(24,123,72),(25,138,88),(26,139,81),(27,140,82),(28,141,83),(29,142,84),(30,143,85),(31,144,86),(32,137,87),(33,73,96),(34,74,89),(35,75,90),(36,76,91),(37,77,92),(38,78,93),(39,79,94),(40,80,95),(41,152,104),(42,145,97),(43,146,98),(44,147,99),(45,148,100),(46,149,101),(47,150,102),(48,151,103),(49,160,112),(50,153,105),(51,154,106),(52,155,107),(53,156,108),(54,157,109),(55,158,110),(56,159,111)], [(1,2,3,4,5,6,7,8),(9,10,11,12,13,14,15,16),(17,18,19,20,21,22,23,24),(25,26,27,28,29,30,31,32),(33,34,35,36,37,38,39,40),(41,42,43,44,45,46,47,48),(49,50,51,52,53,54,55,56),(57,58,59,60,61,62,63,64),(65,66,67,68,69,70,71,72),(73,74,75,76,77,78,79,80),(81,82,83,84,85,86,87,88),(89,90,91,92,93,94,95,96),(97,98,99,100,101,102,103,104),(105,106,107,108,109,110,111,112),(113,114,115,116,117,118,119,120),(121,122,123,124,125,126,127,128),(129,130,131,132,133,134,135,136),(137,138,139,140,141,142,143,144),(145,146,147,148,149,150,151,152),(153,154,155,156,157,158,159,160),(161,162,163,164,165,166,167,168)], [(1,25,113,148,74,65,153),(2,26,114,149,75,66,154),(3,27,115,150,76,67,155),(4,28,116,151,77,68,156),(5,29,117,152,78,69,157),(6,30,118,145,79,70,158),(7,31,119,146,80,71,159),(8,32,120,147,73,72,160),(9,48,37,127,53,168,83),(10,41,38,128,54,161,84),(11,42,39,121,55,162,85),(12,43,40,122,56,163,86),(13,44,33,123,49,164,87),(14,45,34,124,50,165,88),(15,46,35,125,51,166,81),(16,47,36,126,52,167,82),(17,105,132,138,61,100,89),(18,106,133,139,62,101,90),(19,107,134,140,63,102,91),(20,108,135,141,64,103,92),(21,109,136,142,57,104,93),(22,110,129,143,58,97,94),(23,111,130,144,59,98,95),(24,112,131,137,60,99,96)], [(1,153),(2,158),(3,155),(4,160),(5,157),(6,154),(7,159),(8,156),(9,33),(10,38),(11,35),(12,40),(13,37),(14,34),(15,39),(16,36),(17,138),(18,143),(19,140),(20,137),(21,142),(22,139),(23,144),(24,141),(25,65),(26,70),(27,67),(28,72),(29,69),(30,66),(31,71),(32,68),(42,46),(44,48),(49,168),(50,165),(51,162),(52,167),(53,164),(54,161),(55,166),(56,163),(57,93),(58,90),(59,95),(60,92),(61,89),(62,94),(63,91),(64,96),(73,116),(74,113),(75,118),(76,115),(77,120),(78,117),(79,114),(80,119),(81,121),(82,126),(83,123),(84,128),(85,125),(86,122),(87,127),(88,124),(97,101),(99,103),(105,132),(106,129),(107,134),(108,131),(109,136),(110,133),(111,130),(112,135),(145,149),(147,151)]])
102 conjugacy classes
| class | 1 | 2A | 2B | 3A | 3B | 4A | 4B | 4C | 6A | 6B | 6C | 6D | 7A | 7B | 7C | 8A | 8B | 8C | 8D | 12A | 12B | 12C | 12D | 12E | 12F | 14A | 14B | 14C | 21A | ··· | 21F | 24A | 24B | 24C | 24D | 24E | 24F | 24G | 24H | 28A | ··· | 28F | 42A | ··· | 42F | 56A | ··· | 56L | 84A | ··· | 84L | 168A | ··· | 168X |
| order | 1 | 2 | 2 | 3 | 3 | 4 | 4 | 4 | 6 | 6 | 6 | 6 | 7 | 7 | 7 | 8 | 8 | 8 | 8 | 12 | 12 | 12 | 12 | 12 | 12 | 14 | 14 | 14 | 21 | ··· | 21 | 24 | 24 | 24 | 24 | 24 | 24 | 24 | 24 | 28 | ··· | 28 | 42 | ··· | 42 | 56 | ··· | 56 | 84 | ··· | 84 | 168 | ··· | 168 |
| size | 1 | 1 | 14 | 1 | 1 | 1 | 1 | 14 | 1 | 1 | 14 | 14 | 2 | 2 | 2 | 2 | 2 | 14 | 14 | 1 | 1 | 1 | 1 | 14 | 14 | 2 | 2 | 2 | 2 | ··· | 2 | 2 | 2 | 2 | 2 | 14 | 14 | 14 | 14 | 2 | ··· | 2 | 2 | ··· | 2 | 2 | ··· | 2 | 2 | ··· | 2 | 2 | ··· | 2 |
102 irreducible representations
| dim | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 |
| type | + | + | + | + | + | + | ||||||||||||||||
| image | C1 | C2 | C2 | C2 | C3 | C4 | C4 | C6 | C6 | C6 | C12 | C12 | D7 | M4(2) | D14 | C3×D7 | C3×M4(2) | C4×D7 | C6×D7 | C8⋊D7 | C12×D7 | C3×C8⋊D7 |
| kernel | C3×C8⋊D7 | C3×C7⋊C8 | C168 | C12×D7 | C8⋊D7 | C3×Dic7 | C6×D7 | C7⋊C8 | C56 | C4×D7 | Dic7 | D14 | C24 | C21 | C12 | C8 | C7 | C6 | C4 | C3 | C2 | C1 |
| # reps | 1 | 1 | 1 | 1 | 2 | 2 | 2 | 2 | 2 | 2 | 4 | 4 | 3 | 2 | 3 | 6 | 4 | 6 | 6 | 12 | 12 | 24 |
Matrix representation of C3×C8⋊D7 ►in GL2(𝔽13) generated by
| 9 | 0 |
| 0 | 9 |
| 11 | 10 |
| 4 | 2 |
| 0 | 11 |
| 7 | 7 |
| 6 | 8 |
| 7 | 7 |
G:=sub<GL(2,GF(13))| [9,0,0,9],[11,4,10,2],[0,7,11,7],[6,7,8,7] >;
C3×C8⋊D7 in GAP, Magma, Sage, TeX
C_3\times C_8\rtimes D_7
% in TeX
G:=Group("C3xC8:D7"); // GroupNames label
G:=SmallGroup(336,59);
// by ID
G=gap.SmallGroup(336,59);
# by ID
G:=PCGroup([6,-2,-2,-3,-2,-2,-7,313,79,69,10373]);
// Polycyclic
G:=Group<a,b,c,d|a^3=b^8=c^7=d^2=1,a*b=b*a,a*c=c*a,a*d=d*a,b*c=c*b,d*b*d=b^5,d*c*d=c^-1>;
// generators/relations
Export